A general approach to stability in free boundary ...

A general approach to stability in free boundary problems  Claude-Michel Brauner

Universite Bordeaux I, Mathematiques Appliquees de Bordeaux 33405 Talence Cedex, France E-mail: [email protected]

Josephus Hulshof

Mathematical Department of the Leiden University Niels Bohrweg 1, 2333 CA Leiden, The Netherlands E-mail: [email protected] WWW: http://www.wi.leidenuniv.nl/home/hulshof/

Alessandra Lunardi

Dipartimento di Matematica, Universita di Parma Via D'Azeglio 85/A, 43100 Parma, Italy E-mail: [email protected] Abstract

Every solution of a linear elliptic equation on a bounded domain may be considered as an equilibrium of a free boundary problem. The free boundary problem consists of the corresponding parabolic equation on a variable unknown domain with free boundary conditions prescribing both Dirichlet and Neumann data. We establish a rigorous stability analysis of such equilibria, including the construction of stable and unstable manifolds. For this purpose we transform the free boundary problem to a fully nonlinear and nonlocal parabolic problem on a xed domain with fully nonlinear lateral boundary conditions and we develop the general theory for such problems. As an illustration we give two examples, the second being the focussing ame problem in combustion theory.

AMS Subject Classi cation. 35K55, 35K65, 80A25. Key Words and Phrases. Free boundary problems, focussing selfsimilar solutions, lin-

earisation, analytic semigroups, fully nonlinear equations, stable and unstable manifolds, saddle point property.

 We are grateful for the support of the University of Bordeaux-1 and of the HCM-project "Nonlinear PDE-'s" (ERBCHRXCT 940-618). J. Hulshof was partially supported by the TMR network Nonlinear Parabolic Partial Dierential Equations: Methods And Applications, ERBFMRXCT980201. We thank J. Simon for fruitful comments.

1

1 Introduction In this paper we consider free boundary problems of the form

ut = Lu + f; x = (x1 ; : : : ; xN ) 2 t; t > 0;

(1.1)

with free boundary conditions

@u = g on @ : u = 0 and @n (1.2) t Here t is a bounded domain in RN with (free) boundary @ t , L is a uniformly elliptic operator on RN with smooth time independent coecients, and f and g are given smooth functions de ned on the whole of RN . We are interested in the stability properties of equilibria of (1.1),(1.2). We assume that the pair ( ; U ) is a smooth equilibrium, namely

LU + f = 0 in with U = 0 and @U @n = g on @ ;

(1.3)


U = 1 in with U = 0 and @U @n = 1 on @ :

(1.4)

with bounded. We consider (1.1),(1.2) with initial data 0 and u0 close to and U , in a sense to be made precise below. The only structural assumption is that g does not vanish on @ , which we think of as a transversality condition. A simple example is By a suitable change of coordinates, Problem (1.1),(1.2) is reduced to a problem in the xed domain  [0; 1). Then, using a generalisation of methods derived in the framework of travelling waves (see [4],[5],[6]), we get that the local behaviour of solutions near the equilibrium ( ; U ) is determined by the operator L de ned by

Lv = Lv for v 2 D(L) = fv 2 \p>1W 2;p( ); Lv 2 C ( ); Bv = 0 on @ g; where

(1.5)

@v + 1  @g ? @ 2 U v: Bv = @n g @n @n2

(1.6)

Bw = G(w; Dw) on @ ;

(1.9)

This boundary operator is strongly reminiscent of a `Hadamard formula' derived in [19] in the context of shape optimisation problems. More precisely, we set u = U + 
rU + w; (1.7) where  measures the local dierence between the free boundary @ t and the xed boundary @ , and will be de ned in Section 2. We emphasise that the unknown  does not appear in the nal formulation: Problem (1.1),(1.2) is rewritten for w as the fully nonlinear problem wt = Lw + F (w; Dw; D2 w) on ; t > 0; (1.8) with boundary conditions also fully nonlinear, where G only depends on w and its tangential derivatives. The price to be paid for changing the free boundary problem to a xed boundary problem are the nonlocal and fully nonlinear terms F and G. As initial datum for w we take w(x; 0) = w0 (x) for x 2 , with w0 2 C 2+ ( ) close to 0 in this norm, satisfying the compatibility condition Bw0 = G (w0 ; Dw0 ) on @ . This 2

will correspond to initial data 0 and u0 close (in the new coordinates) to and U in the C 2+-norm, satisfying the two compatibility conditions 0 u0 = 0 and @u @n = g on @ 0 :

(1.10)

Thus, near an equilibrium, the free boundary problem is equivalent to a problem on a xed domain: the solution of (1.1),(1.2) with initial data u0 and 0 can be retrieved from the solution of (1.8),(1.9) with initial datum w0 . Our rst result concerns the local existence.

Theorem 1.1 For every T > 0 there are r,  > 0 such that Problem (1.8)-(1.9) has a solution w 2 C 1+=2;2+ ([0; T ]  ) if kw0 kC 2+ ( )  . Moreover w is the unique solution in B (0; r)  C 1+=2;2+ ([0; T ]  ). The reformulation of (1.1),(1.2) as the fully nonlinear problem (1.8),(1.9) with initial datum w0 also enables us to perform a stability analysis of the free boundary problem. The stability question for the original equilibrium (1.3) is rephrased as the stability of w = 0 for Problem (1.8),(1.9).

Theorem 1.2 If all elements of the spectrum (L) of the operator L have negative real

part then w = 0 is a stable equilibrium of Problem (1.8),(1.9). If (L) contains elements with positive real part then w = 0 is unstable.

For the description of the stable and unstable manifolds we need the spectral projections P + and P ? associated to the subsets of (L) with positive and negative real parts.

Theorem 1.3 Assume that (L) contains elements with positive real part. (i) There exists a unique local unstable manifold of the form

' : B (0; r0 )  P +(C 2+ ( )) ! (I ? P +)(C 2+( )); ' Lipschitz continuous, dierentiable at 0 with '0 (0) = 0. (ii) There exist a unique local stable manifold of the form

: B (0; r1 )  fw0 2 P ? (C 2+ ( )) : Bw0 = G(w0 (
))g ! (I ? P ? )(C 2+ ( )); Lipschitz continuous, dierentiable at 0 with 0 (0) = 0.

We note that in the special case that

(L) \ iR = ;;

(1.11)

Theorem 1.3 is a generalisation of the classical saddle point theorem. Our investigations are motivated by the mathematical modelling of combustion [16], where typically at the free boundary separating the fresh and the burnt regions one encounters jump conditions involving the normal derivative @u=@n. In particular we consider the model problem (see the survey [21])

@u = 1 on @ : ut =
u in t; u = 0 and @n t

(1.12)

In the case of bounded domains, (1.12) does not allow nontrivial equilibria. It is called a focussing problem because typically the domain t vanishes in nite time, a behaviour 3

exhibited by selfsimilar solutions. Transforming to selfsimilar variables the problem can be rewritten as ut = Lu =
u ? 12 x
ru + 12 u; x 2 t (1.13) with the same boundary conditions. The stability analysis of the equilibrium corresponding to the selfsimilar pro le ts in the framework presented above. The paper is organised as follows. In Section 2 we describe the general method to transform (1.1),(1.2) into (1.8),(1.9). In fact, we consider a slightly more general problem, namely ut = Lu + f; x = (x1 ; : : : ; xN ) 2 t; t > 0; (1.14) with @u = g on @ : (1.15) u = g1 and @n 2 t In Section 3 we give the extension of the local existence and saddle point theorems in [17] to the case of (1.8),(1.9). This is needed because unlike in [3], where we had a linear boundary condition, we now arrive at the fully nonlinear boundary condition (1.9) and therefore a signi cant part of this paper will deal with adapting the functional analytic framework to this case. The proofs are based on xed point arguments and rely on results for the asymptotic behaviour of linear inhomogeneous problems which we give in the appendix. In Section 4 we describe the application to the two examples mentioned above.

2 The general procedure for linearisation

We consider Problem (1.1),(1.15) from the introduction, where L is a uniformly elliptic operator with smooth coecients, and f , g1 and g2 are given smooth functions de ned on the whole of RN . We assume that the pair ( ; U ), with bounded, @ and U smooth, is an equilibrium:

LU + f = 0 in with U = g1 and @U @n = g2 on @ :

(2.1)

@g1 ? g 6= 0 at @ : @n 2

(2.2)

The structural assumption we need is that

Thus, obstacle type problem are excluded. The method consists in  xing the domain by transforming to new independent variables  2 and  ;  linearising around U in the new variables;  rewriting the problem as a fully nonlinear problem in .

2.1 Fixing the domain

We transform the problem on the variable domain t (space variable x, time variable t) to a problem on the xed domain (space variable  , time variable  ). The normal to @ t being denoted by n, we will use  for the normal on @ . For  > 0 we de ne a map X : @  [?; ] ! RN ; X (; r) =  + r (): (2.3) If  is suciently small, then (2.3) de nes a bijection to a compact neighbourhood N (@ ) of @ . In N (@ ) every  can be written in a unique way as  = X (; r) with  2 @ and r 2 [?; ]. For convenience we shall write 0 for . Clearly, if  2 @ then  = 0 . 4

We will look for t close to in some time interval I in the sense that @ t will be given by @ t = fx = 0 + s(0; t) (0 ); 0 2 @ g; (2.4) where s : @  I ! [?; ] is a smooth function which is an unknown of the problem. In what follows it will be convenient to write, for  2 @ , (; t) = s(; t) ( ):

(2.5)

We extend this eld to the whole of RN by setting 8 >


(2.6)

Here  : RN ! [0; 1] is a smooth molli er which is equal to 1 near @ and has compact support in the interior of N (@ ). Thus, the eld  is localised near @ . It contains all the information about the dierence between the xed boundary @ and the free boundary @ t . Now it is clear how to transform the problem on the variable domain t to a problem on the xed domain . We de ne a (bijective) coordinate transformation

x =  + (;  ); t = : From (2.7) and (2.5) we see that rx and where

@ @t

(2.7)

transform as

rx = (I + A)?1 r ; @t@ = D^  = @@ ? @@
(I + A)?1 r ;

(2.8)

A = A(;  ) = r (;  )

(2.9) is the column gradient of , i.e. the transposed Jacobian matrix of  (with respect to the -variables only). The normal n on @ t is related to the normal  on @ by + A)?1  : n = j((II + A)?1 j

(2.10)

2.2 Expansion in the new variables

The transformation of t to also acts on the equilibrium U itself and this has to be taken into account when we expand u near U in the new variables. It is convenient to extend U to a smooth function de ned in the whole of RN . The extension of U outside

will not appear in the nal formulation. Observing that

U ( + (;  )) = U () + (r U ())
(;  ) + R(; (;  )); (2.11) where here and in what follows R(; ) is a nonspeci ed smooth function bounded by a multiple of jj2 , we are led to look for a solution the form u(x; t) = u^(;  ) = U () + (r U ())
(;  ) + w(;  ):

(2.12)

This non-standard formula is crucial: it will allow us to rewrite the free boundary problem as a problem for w. First we rewrite the action of L on u as (Lu)(x; t) = (L^u^)(;  ) = (L^0 u^)(;  ) + (L^1 u^)(;  ) + (L^r2 u^)(;  ); (2.13) 5

where L^0 is the original operator L with x replaced by  and L^1 is an operator with coecients linear in  and its rst and second order  -derivatives. The remainder term L^r2 has coecients bounded by a multiple of jj2 + jD j2 + jD2j2 . This is easily seen from the expansion (I + A)?1 = I ? A + A2 (I + A)?1 ; A = r :

(2.14)

Likewise we get an expansion for the time derivative which reads @u(x; t) = D^ u^(;  ) = @ u^ ? @ 
r u^ + @ 
A(I + A)?1 r u^:

@t



@

@





@

Equation (1.1) for u thus transforms into an equation for u^, (D ? L^)(^u) = f ( + (;  )) = f ( ) + (r f ( ))
(;  ) + R(; (;  ));

(2.15) (2.16)

in which we substitute (2.12). Next we expand the resulting equation using (2.13) and (2.15). We observe beforehand that the terms of zero and rst order in (;  ) and its derivatives coming from the rst two terms of (2.12) must be equal on both sides and arrive at @w ? L^ w =  @ 
r + L^ w +  ? @ 
A(I + A)?1 r + L^r w (2.17)

@

0

@



1

@



2

    + @@
r + L^1 (
r U ) + ? @@
A(I + A)?1 r + L^r2 (U + 
r U ) + R(; (;  )):

The linear part (the left hand side) is now in the form we want, but the higher order terms contain derivatives of both  and w, including @@ . Equation (2.17) is of the type @w ? L^ w = F (; w; Dw; D2 w; ; D; D2 ) + @ 
F (; Dw; ; D); (2.18)

1 @ 0 @ 2 with F1 consisting of second and higher order terms and F2 consisting of rst and higher order terms in the w; -dependent arguments.

To eliminate  and its derivatives from (2.18) we have to transform the free boundary conditions.

2.3 Expansion on the boundary: new boundary conditions

At the boundary the expansion (2.12) reduces to, using (1.15) and (2.5),

g1 ( + s(;  ) ()) = g1 () + s(;  )g2 () + w(;  );  2 @ :

(2.19)

Expanding the left hand side this gives

  1 ( ) ? g ( ) + R(; s(;  )) = w(;  ); s(;  ) @g 2 @

(2.20)

where R(; s) is as before a nonspeci ed smooth function bounded by a multiple of s2 . In view of the structural assumption (2.2) we may invert this equation for w small to obtain s(;  ) = @g1 w(;  ) + R(; w(;  )); (2.21) (  ) ? g (  ) 2 @ where R(; w) is again smooth and bounded by a multiple of w2 . In the special case that g1 is constant this term is zero. Note that w and s (or ) are of the same (small) size. More importantly, (2.20) will allow us to decouple the system and obtain a problem for w only. 6

To transform the second free boundary condition in (1.15) we have to expand (2.10) using (2.14). We nd, since

A = (r ) = r s = rtang  s

(2.22)

(not to be confused with (D ) = 0), whence A
 = 0,

n =  ? r s + R(A):

(2.23)

The vector valued function R(A) = R(; s; r s) is bounded by a multiple of jjAjj2 and hence, in view of (2.6) and (2.9), by a multiple of s2 + jr sj2 . For the normal derivative we then have

@ = n
r = 
r ? r s
r + r s
A(I + A)?1 r + R(A)
(I + A)?1 r ; x       @n

which we can rewrite as

@ = @ ? rtang s
rtang + B (; s; r s)r ;    @n @ 

(2.24)

with B a smooth matrix valued function bounded by a multiple of s2 + jr sj2 . Applying (2.24) to (2.12), expanding g2 ( + (;  )) as usual, we arrive at, omitting the subscripts  ,

@w + s @ 2 U ? @g2  ? rtang s
rtang g = B (; s; rs)rw + R(; s; rs); (2.25) 1 @ @ 2 @ with a slightly dierent matrix B . The derivation of this boundary conditon for w is independent of the partial dierential equation to be satis ed in the interior. What we have done here is related to methods in domain optimisation problems where solutions of boundary value problems are dierentiated with respect to the domain. See [18, 19, 10, 8].

2.4 The fully nonlinear problem

We recall that we have derived (2.18) for  in and equations (2.20) and (2.25) for  on the boundary @ . Note that (2.20) is equivalent to (2.21). Now we can eliminate the free boundary terms s,  and their derivatives from the problem. We take the restriction of equation (2.18) to the boundary @ (where (;  ) = s(;  ) ()) and rewrite it as

@w ? L^ w = F (; w; Dw; D2 w; s; Ds; D2 s) + @s F (; Dw; s; Ds); (2.26) 1 @ 0 @ 2 where now F1 and F2 are considered to be depending on s rather than on . In (2.26) we use (2.21) to express s, Ds and D2 s in terms of w, Dw, D2 w and  . The important point

is the elimination of  to obtain

@s @ .

For this we use (2.20) which we rst dierentiate with respect to

@w ?  @g1 ? g  @s = R (; s) @s = R~ (; w) @s : (2.27) s @ @ 2 @ @ @ The partial derivative Rs (; s) is a smooth function bounded by a multiple of s and thus, using (2.21) again, R~ (; w) is smooth and bounded by a multiple of w. Using (2.27) to eliminate 

@w @

we obtain

@g1 ? g + R~ (; w) @s ? L^ w = F (; w; Dw; D2 w) + @s F (; w; Dw; D2 w); (2.28) 1 @ 2 @ 0 @ 2 7

where F1 and F2 are the same as before, with the s-dependence absorbed in w-dependence. Therefore, provided w, Dw and D2 w are small, @s = L^0w + F1 (; w; Dw; D2 w) (2.29) @ @g@1 ? g2 + R~ (; w) ? F2 (; w; Dw; D2 w) : We emphasise that (2.29) holds for  at the boundary @ only. Returning to (2.18) and using (2.6) we have for  2 \ N (@ ) (where ( ) is supported) that the equation for w becomes of the form

@w ? L^ w = F (; w(;  ); Dw(;  ); D2 w(;  ); w(0 ;  ); Dw(0 ;  ); D2 w(0 ;  )); (2.30) @ 0 with F smooth, quadratic in the w-dependent variables. We recall that  0 2 @ is the projection of  2 N (@ ) on the boundary, see the beginning of Section 2.1. Thus the

right hand side of (2.31) is fully nonlinear and nonlocal. Note that the right hand side, which contains the nonlocal terms, may be nonzero only for  2 \ N (@ ), where ( ) is supported. Elsewhere in it vanishes identically. Therefore, the nal equation for w is 8 > > >
> > : 0 for  2 n N (@ );   0: The boundary condition to be satis ed by w follows directly from (2.20), (2.21) and

(2.25). It comes out as

w  @ 2U ? @g2  ? rtang  w 
rtang g = G (; w; Dw); (2.32) Bw = @w + 1 @g1 ? g @ @g@1 ? g2 @ 2 @ 2 @ In the special case that g1  0 and g2 = g the boundary operator de ned by (2.32) reduces to Bw de ned by (1.6). Now we have found a fully nonlinear problem for w, namely (2.31),(2.32), with initial datum w(; 0) = w0 ( ) determined by 0 and u0 via (2.4) and (2.12). Since u0 is assumed to satisfy the boundary conditions (1.15) at t = 0, it follows that w0 satis es Bw0 = G (
; w0 ; Dw0 ) at @ . We shall solve this initial boundary value problem for w provided w0 is suciently small by means of the general results in Section 3.

3 General theory Problem (2.31),(2.32) is of the type 8 >
:

Bw = G(w(
; t))();  2 @ ;

(3.1)

with initial condition given by

w(; 0) = w0 ();  2 ;

(3.2)

where is a bounded open set in RN with regular boundary @ , F and G are regular functions de ned in a neighbourhood of 0 in C 2 ( ) with values in C ( ) and C 1 (@ ) respectively and L and B are linear dierential operators with regular coecients. Moreover,

F (0) = 0; F 0 (0) = 0; G(0) = 0; G0(0) = 0; 8

so that w  0 is a solution of Problem (3.1) and the linearisation of (3.1) around the null solution is wt = Lw with boundary condition Bw = 0. Below we state precise regularity assumptions for the local existence and uniqueness of a regular solution to (3.1),(3.2) and the construction of the stable and unstable manifolds of the null solution. Such assumptions are easily seen to be satis ed in the case of Problem (2.31),(2.32). H1. is a bounded open set in RN with C 2+ boundary, 0 <  < 1. H2. F : B (0; R)  C 2( ) ! C ( ) is continuously dierentiable with Lipschitz continuous derivative, F (0) = 0; F 0 (0) = 0 and the restriction of F to B (0; R)  C 2+ ( ) has values in C  ( ) and is continuously dierentiable; G : B (0; R)  C 1 ( ) ! C (@ ) is continuously dierentiable with Lipschitz continuous derivative, G(0) = 0; G0 (0) = 0 and the restriction of G to B (0; R)  C 2+ ( ) has values in C 1+ (@ ) and is continuously dierentiable too. H3. L = aij Dij + biDi + c is a uniformly elliptic operator with coecients aij , bi, c in C ( ) and B = i Di + is a nontangential operator with coecients i , in C 1+ (@ ). H4. w0 2 C 2+ ( ) satis es the compatibility condition

Bw0 = G(w0 (
))();  2 @ :

(3.3)

A local existence and uniqueness result for Problem (3.1),(3.2) may be shown using a standard linearisation procedure, which works also in the fully nonlinear case thanks to optimal regularity results and estimates for linear problems. It gives existence of the solution for arbitrarily long time intervals, provided the initial datum is small enough. This may be seen as continuous dependence of the solution on the initial datum at w0 = 0. We shall use the functional spaces C ;=2 (  I ), C 2+;1+=2 (  I ), C 1+;1=2+=2 (@  I ), I being a real interval, with the usual meanings and norms. We recall that a function w belongs to C 2+;1+=2 (  I ) if and only if t ! w(
; t) is in C 1+=2 (I ; C ( )) \ B (I ; C 2+ ( )) (B stands for bounded) and there is K > 0 such that

kwkC 2+;1+=2 ( I )  K(kwkC 1+=2 (I ;C ( )) + kwkB(I ;C 2+ ( )) ): Similarly, v 2 C 1+;1=2+=2 (@  I ) if and only if t ! v(
; t) is in C 1=2+=2 (I ; C (@ )) \ B (I ; C 1+ (@ )) and there is C > 0 (independent of I ) such that

kvkC 1+;1=2+=2 (@ I )  C (kvkC 1=2+=2 (I ;C (@ )) + kvkB(I ;C 1+ (@ )) ):

Theorem 3.1 Under the above assumptions, for every T > 0 there are r,  > 0 such that (3.1),(3.2) has a solution w 2 C 2+;1+=2 (  [0; T ]) if kw0 kC 2+ ( )  . Moreover w is the unique solution in B (0; r)  C 2+;1+=2 (  [0; T ]). Proof | Let 0 < r  R, and set K (r) = supfkF 0 (')kL(C 2+ ( );C ( )) : ' 2 B (0; r)  C 2+( )g; H (r) = supfkG0 (')kL(C 2+ ( );C 1+(@ )) : ' 2 B (0; r)  C 2+( )g: Since F 0 (0) = 0 and G0 (0) = 0, K (r) and H (r) go to 0 as r ! 0. Let L > 0 be such that, for all ', 2 B (0; r)  C 2 ( ) with small r, kF 0 (') ? F 0( )kL(C 2 ( );C ( ))  Lk' ? kC 2 ( ) ;

kG0 (') ? G0( )kL(C 1 ( );C (@ ))  Lk' ? kC 1 ( ) : 9

For every 0  s  t  T and for every w 2 B (0; r)  C 2+;1+=2 (  [0; T ]) with r so small that K (r); H (r) < 1, we have

kF (w(
; t))kC  ( )  K (r)kw(
; t)kC 2+ ( ); kF (w(
; t)) ? F (w(
; s))kC ( )  and similarly

Lrkw(
; t) ? w(
; s)kC 2 ( )  Lrjt ? sj=2 kwkC 2+;1+=2 ( [0;T ]);

kG(w(
; t))kC 1+ (@ )  H (r)kw(
; t)kC 2+ ( )  H (r)kwkC 2+;1+=2 ( [0;T ]); kG(w(
; t)) ? G(w(
; s))kC (@ )  Lrkw(
; t) ? w(
; s)kC 1 ( )  Lrjt ? sj1=2+=2 kwkC 2+;1+=2 ( [0;T ]): Therefore, (; t) ! F (w(
; t))( ) belongs to C ;=2 ( [0; T ]), (; t) ! G(w(
; t))( ) belongs to C 1+;1=2+=2 (@  [0; T ]) and

kF (w)kC ;=2 ( [0;T ])  (K (r) + Lr)kwkC 2+;1+=2 ( [0;T ]); kG(w)kC 1+;1=2+=2 (@ [0;T ])  C (2H (r) + Lr)kwkC 2+;1+=2 ( [0;T ]): So, if kw0 kC 2+ ( ) is small enough, we de ne a nonlinear map ? : fw 2 B (0; r)  C 2+;1+=2 (  [0; T ]) : w(
; 0) = w0 g ! C 2+;1+=2 (  [0; T ]); by ?w = v, where v is the solution of

8 vt (x; t) = Lv + F (w(
; t))(x); > > > > > < > > > > > :

0  t  T; x 2 ;

Bv = G(w(
; t))(x); 0  t  T; x 2 @ ; v(x; 0) = w0 (x):

Actually, thanks to the compatibility condition Bw0 = G(w0 ) and the regularity of F (w) and G(w), the range of ? is contained in C 2+;1+=2 (  [0; T ]). Moreover, there is C = C (T ) > 0, independent of r, such that (see e.g. [15, Thm. 5.3] or [17, Thm. 5.1.20])

kvkC 2+;1+=2 ( [0;T ])  C (kw0 kC 2+( ) + kF (w)kC ;=2 ( [0;T ]) + kG(w)kC 1+;1=2+=2 (@ [0;T ])); so that

k?(w)kC 2+;1+=2 ( [0;T ])  C (kw0 kC 2+ ( ) + (K (r) + Lr + C(2H (r) + Lr))kwkC 2+;1+=2 ( [0;T ])):

Therefore, if r is so small that

C (K (r) + Lr + C(2H (r) + Lr))  1=2; and w0 is so small that

kw0 kC 2+ ( )  Cr=2; 10

(3.4)

? maps the ball B (0; r) into itself. Let us check that ? is a 1=2-contraction. Let w1 , w2 2 B (0; r), wi(
; 0) = w0 . Writing wi(
; t) = wi(t), i = 1; 2, we have

k?w1 ? ?w2 kC 2+;1+=2 ( [0;T ])  C (kF (w1 ) ? F (w2 )kC ;=2 ( [0;T ]) + kG(w1 ) ? G(w2)kC 1+;1=2+=2 (@ [0;T ])); and, arguing as above, for 0  t  T , kF (w1 (t)) ? F (w2 (t))kC  ( )  K (r)kw1 (t) ? w2 (t)kC 2+ ( )  K (r)kw1 ? w2 kC 2+;1+=2 ( [0;T ]);

kG(w1(t)) ? G(w2 (t))kC 1+ (@ )  H (r)kw1 (t) ? w2 (t)kC 2+ ( )  H (r)kw1 ? w2kC 2+;1+=2 ( [0;T ]); while for 0  s  t  T kF (w1 (t)) ? F (w2 (t)) ? F (w1(s)) ? F (w2 (s))kC ( ) =

Z 1

F 0 (w1 (t) + (1 ?  )w2 (t)(w1 (
; t) ? w2 (t))

0



?F 0(w1 (s) + (1 ? )w2 (s))(w1 (s) ? w2(s))d



Z 1

0

C ( )

k(F 0 (w1 (t) + (1 ? )w2 (t) ? F 0 (w1 (s) + (1 ? )w2 (s))(w1 (t) ? w2 (t))dkC ( )

Z 1

+ kF 0 (w1 (s))(w1 (t) ? w2 (t) ? w1 (s) + w2 (s))kC ( ) 0

 L2 (kw1 (t) ? w1 (s)kC 2 ( ) + kw2 (t) ? w2(s)kC 2 ( ) )kw1 (t) ? w2 (t)kC 2 ( ) +Lrkw1 (t) ? w2 (t) ? w1 (s) + w2 (s))kC 2 ( )

 2Lr(t ? s)=2 kw1 ? w2 kC 2+;1+=2 ( [0;T ]); and similarly

kG(w1 (
; t)) ? G(w2(
; t)) ? G(w1(
; s)) ? G(w2 (
; s))kC (@ )  2Lr(t ? s)1=2+=2 kw1 ? w2 kC 2+;1+=2 ( [0;T ]): Therefore,

k?w1 ? ?w2kC 2+;1+=2 ( [0;T ])  C (K (r) + Lr + C (2H (r) + Lr))kw1 ? w2 kC 2+;1+=2 ( [0;T ])  12 kw1 ? w2kC 2+;1+=2 ( [0;T ]);

the last inequality being a consequence of (3.4). The statement follows. 11

As in Section 1 we de ne the realisation of L with homogeneous boundary conditions in X = C ( ) by

D(L) = fv 2 \p1W 2;p( ) : Lv 2 X; Bv = 0 in @ g; (3.5) Lv = Lv; v 2 D(L): We note that, due to [20], L is a sectorial operator and since is bounded, the resolvent (I ? L)?1 is a compact operator for every  in the resolvent set (L). Therefore (L)

consists of a sequence of isolated eigenvalues. We shall state the principle of linearised stability in terms of the spectrum (L) of L.

Theorem 3.2 Let satisfy assumption H1 and let F , G, L, B satisfy H2, H3. (i) If all the elements of (L) have negative real part then w = 0 is a stable equilibrium of Problem (3.1) with respect to the C 2+ ( ) norm. More precisely, for every ! 2 (0; maxfRe  :  2 (L)g), there are C , r > 0 such that for every w0 satisfying H4 and kw0 kC 2+ ( )  r, the solution of (3.1) with initial datum w0 exists in the large and satis es kw(
; t)kC 2+ ( )  Ce?!t kw0 kC 2+( ); t  0: (ii) If (L) contains elements with positive real part then w = 0 is unstable in C 2+ ( ).

Proof | The proof of statement (i) follows the proof of Theorem 9.1.2 of [17], replacing the space Y used in [17] by the space of functions w such that (; t) ! e?!t w(; t) 2 C 2+;1+=2 (  [0; 1)). This is possible thanks to Theorem 0.1 in the appendix. The proof of (ii) follows the proof of Theorem 9.1.3 of [17], replacing the space

C ((?1; 0]; D; !) used in [17] by the space of the functions v such that (; t) ! e? t v(; t) 2 C 2+;1+=2 (  (?1; 0]), with any 2 (0; minfRe  :  2 (L); Re  > 0g).

In the unstable case we shall construct the stable and unstable manifolds arguing as in Theorems 9.1.3 and 9.1.4 of [17]. To do this we shall use asymptotic behaviour results for forward and backward linear problems, whose precise statements and proofs we defer to the Appendix. We need some notation. We denote by + (L), 0 (L) and ? (L) the subsets of (L) respectively consisting of elements with positive, zero and negative real parts. Since (L) is discrete, + (L), 0 (L) and ? (L) are spectral sets. Let P + and P ? be the spectral projections associated to + (L) and ? (L).

Theorem 3.3 Let satisfy assumption H 1 and let F , G, L, B satisfy H 2, H 3. (i) Assume that + (L) = 6 ; and x ! 2 (0; minfRe :  2 +(L)g). Then there exist R0 ,

r0 > 0 and a Lipschitz continuous function ' : B (0; r0 )  P +(C ( )) = P +(C 2+ ( )) ! (I ? P +)(C 2+ ( )); dierentiable at 0 with '0 (0) = 0, such that for every w0 belonging to the graph of ', Problem (3.1) has a unique backward solution v such that v~ de ned by v~(; t) = e?!t v(; t), belongs to C 2+;1+=2 (  (?1; 0]) and satis es kv~kC 2+;1+=2 ( (?1;0])  R0 : (3.6)

Moreover, for every !0 2 (0; minfRe :  2 + (L)g) we have (; t) ! e?!0 t v(; t) 2 C 2+;1+=2 (  (?1; 0]). Conversely, if Problem (3.1) has a backward solution v which satis es (3.6) and kP + v(0)k  r0 , then v(0) 2 graph '.

12

(ii) Fix ! 2 (0; ? maxfRe :  2 ? (L)g). Then there exist R1 , r1 > 0 and a Lipschitz continuous function

: B (0; r1 )  fw0 2 P ? (C 2+ ( )) : Bw0 = G(w0 (
))g ! (I ? P ? )(C 2+ ( )); dierentiable at 0 with 0 (0) = 0, such that for every w0 belonging to the graph of , Problem (3.1) has a unique solution w such that w~ de ned by w~ (; t) = e!t w(; t) belongs to C 2+;1+=2 (  [0; 1)) and

kw~kC 2+;1+=2 ( [0;1))  R1:

(3.7)

Moreover, for every !0 2 (0; ? maxfRe :  2 ? (L)g) we have (; t) ! e!0 t w(; t) 2 C 2+;1+=2 (  [0; 1)). Conversely, if Problem (3.1) has a forward solution w which satis es (3.7) and kP ? w(
; 0)kC 2+ ( )  r1 , then w(
; 0) 2 graph .

Proof | The proof closely follows the proof of Theorems 9.1.3 and 9.1.4 of [17] using as

main tools Theorems 0.1 and 0.2. Of course the spaces used in Theorems 9.1.3 and 9.1.4 of [17] have to be replaced by the spaces of the functions w such that (; t) ! e?!t w(; t) 2 C 2+;1+=2 (  (?1; 0]) and (; t) ! e!t w(; t) 2 C 2+;1+=2 (  (0; 1)). Note that Theorems 9.1.3 and 9.1.4 of [17] are stated under the assumption that the spectrum of L does not intersect the imaginary axis. Here, since we only have a discrete spectrum, this assumption may be dropped.

4 Applications We recall that we have transformed Problem (1.14),(1.15) with initial data u0 and @ 0 to the fully nonlinear Problem (3.1) with initial condition (3.2), where w0 is given by

w0 () = u0( + 0()) ? U () ? r U ()
0 (): (4.1) Thus w0 depends on u0 as well as on @ 0 , which determines 0 ( ) = ( )s( 0 ; 0) ( 0 ) through (2.4) and (2.6) with t =  = 0. We solved Problem (3.1),(3.2) under the assumption that w0 is small in C 2+ ( ) and satis es the compatibility condition (3.3). In view of the smoothness of the data U and @ , the smallness condition on w0 is equivalent to the assumption that @ 0 is C 2+close to @ and that u0 is C 2+ -close to U (through the inverse of the C 2+ -bijection  2 !  +0() 2 0). Moreover, the compatibility condition (3.3) is satis ed provided (1.10) holds, which, for Problem (1.14),(1.15), reads

0 u0 = g1 and @u @n = g2 on @ 0 :

(4.2)

Having solved Problem (3.1),(3.2), we recover the solution (u(
; t); t ) of the original free boundary problem, using (2.19) rewritten as (2.21) to express s in terms of w, then de ning the free boundary @ t by (2.4), returning to the original space variables by (2.7) and nally recovering u(x; t) by (2.12). We consider the two speci c examples mentioned in the introduction to illustrate the ideas and the theory developed in Sections 2 and 3. In these two cases we are able to verify the spectral hypotheses of Theorems 1.2 and 1.3.

13

4.1 A simple example

The rst example we discuss is the case that, in the notation of (1.1),(1.2), L =
, f = ?1 and g = 1, i.e.

@u = 1 on @ : ut =
u ? 1 in t ; u = 0 and @n t

(4.3)

The unique (up to translations) negative radial equilibrium solution is

2 U (x) = j2xNj ? N2 ; = BN = fx 2 RN : jxj < N g: The operator L : D(L) ! C ( ) is de ned by Lv =
v for v 2 D(L) = fv 2 \p>1W 2;p( );
v 2 C ( ); Bv = 0 on @ g;

with

@v ? 1 v: Bv = @n N

(4.4) (4.5) (4.6)

It is a standard exercise in spectral analysis to show that the spectrum (L) consists entirely of real eigenvalues which may be found, using separation of variables, from the radial ordinary dierential equation 00 + N ? 1 0 ? n(n + N ? 2) =  ; (r)  rn as r ! 0; (4.7)

r

r2

with boundary condition

0 (N ) = 1 (N ):

N

(4.8)

The angular part of the eigenfunction is then a harmonic polynomial of degree n. We have for each n = 0; 1; 2; : : : a sequence of eigenvalues

(1n) > (2n) > (3n) >


# ?1;

(4.9)

with corresponding solutions 1(n) ; 2(n) ; 3(n) ; : : :. Each j(n) has exactly j ? 1 sign changes in the interval (0; N ). It follows from Sturm's Comparison Theorem that the double sequence (jn) is also decreasing in j . In view of the translation invariance we may expect to have zero as an eigenvalue. @U , i = 1; : : : ; N , are easily seen to be eigenfunctions with eigenvalue Indeed, the functions @x i (1) 0. They correspond to n = j = 1, i.e. (1) 1 = 0 with 1 (r) = r. As a consequence of the monotonicity properties of the (jn) we then have that (0) 1 > 0, so that U is unstable, (0) (n) thanks to Theorem 1.2. We note that except for 1 > 0 and (1) 1 = 0, all the j are negative. To see this it is sucient to show that (0) 2 < 0. This is clear because a solution of (4.7) with n = 0 and   0 cannot change sign while 2(0) (r) does.

4.2 The focussing problem The study of (1.12), i.e.

@u = 1 on @ ; ut =
u in t; u = 0 and @n t

(4.10)

is motivated by the mathematical modelling of combustion [16, 21]. Problem (4.10) can be seen as the physical high activation limit of the regularising problems ut =
u + f(u), where f has support in a small interval [?; 0] and is of the form f(s) = f1 (s=)=. In 14

[9] this regularisation is used to prove an existence result for (4.10) under appropriate conditions on the initial data. The same technique has previously been used in [1] for stationary problems and especially for the travelling wave case. As far as we know the uniqueness question for Problem (4.10) has not been settled yet allthough it is clear that additional assumptions have to be made. Examples of nonuniqueness may be constructed even in dimension N = 1: taking initial data with an internal zero we can choose between a solution for which the free domain splits up in two domains or a solution for which the zero disappears instantaneously. Under additional assumptions avoiding such counterexamples, wellposedness results have been obtained in [14] and [11] for radial and one-dimensional problems. Equilibria of (4.10) with bounded do not exist. Thus solutions of (4.10) cannot be expected to stabilise. Typically the domain t vanishes in nite time, a behaviour exhibited by selfsimilar solutions of the form

p p u(x; t) = T ? t f () ;  = p jxj ; t = fjxj < b T ? tg T ?t

(4.11)

with f () satisfying f 00() + N ? 1 f 0() + 21 f = 12 f 0 for 0    b ; f 0(0) = f (b) = 0 ; f 0(b) = 1 : (4.12) This uniquely determines the \free boundary" b = bN for which exactly one solution f of (4.12) with f < 0 on [0; b) exists, and yields a similarity solution of (4.10) which \focusses" in the origin at t = T . In [14] and [11] it is shown that radial solutions focus at the origin in nite time and that they are asymptotically selfsimilar. In one dimension the latter is also true for nonsymmetric initial data, in which case the asymptotic pro le is symmetric but the focussing point does not have to be the origin. A natural question is to ask whether in every space dimension all solutions are asymptotically selfsimilar. However, posed in such generality, the answer is negative since it was proved in [11] that radial annular solutions focus on a sphere, provided the inner hole does not shrink to a point. On the other hand, since it is tempting to speculate, we conjecture that such a behaviour is unstable under nonradial perturbations causing the domain to change its topology. Further speculations, in particular on necessary and sucient conditions on the initial data in terms of e.g. the shape of the domain for asymptotic selfsimilarity, are left to the reader. Here we restrict ourselves to a local armative answer given by the local (linearised) stability analysis of (4.11). To perform this analysis we follow [14] and [11] and transform the problem to selfsimilar variables (4.13) x~ = x 1 ; t~ = ? log(T ? t); u~(~x; t~) = u(x; t)1 ; ~ t~ = fx~ : x 2 tg: (T ? t) 2 (T ? t) 2 Omitting the tildes, we arrive at (4.14) ut =
u ? 12 x
ru + 21 u; x 2 t with boundary conditions @u = 1 on @ : u = 0; @n (4.15) t The selfsimilar solution (4.11) is transformed by (4.13) into an equilibrium

U (x) = f (jxj); = Bb = fx 2 RN : jxj < bg; for (4.14),(4.15). 15

(4.16)

In a previous work with C. Schmidt-Laine [3] we have established a saddle point property of (4.11) in the class of radial solutions. The local analysis in [3] is done by scaling the radial r-variable to x the free boundary and using a splitting which is essentially equivalent to (1.7). In that case the boundary condition (1.9) comes out to be linear, i.e. G (w) = 0. As announced in [3] the scaling trick can also be used for nonradial perturbations of (4.16) but it leads to singularities at r = 0 in the higher order terms. To avoid this problem one has to localise the transformation which xes the free boundary, and this may be done using a molli er. Rather than presenting such an adaptation which works for radial equilibria only, we have chosen to develop the general approach of Section 2, which does not require any symmetry and is basically a localisation of the method in [4, 5, 6]. The application to the focussing problem and the spectral analysis below extend the result of [3] and give a saddle point property of (4.16) in the class of all (radial and nonradial) solutions. We cannot expect (4.16) to be stable because the original problem is invariant under translations in x and t. Thus if we apply a small shift to (4.16), we obtain another selfsimilar solution which is transformed by (4.13) into a solution which starts close to (4.16) but moves away from it. The unstable manifold of (4.16) must therefore contain the images under (4.13) of shifts in space and time of (4.16). These are given by  x ? 1e 12 t ; 1 + 2 et U p 2 et + 1

p

(4.17)

where 1 2 RN and 2 2 R. As a consequence we have that L : D(L) ! C (Bb ) de ned by Lv =
v ? 12 x
rv + 12 v; (4.18) for (4.19) v 2 D(L) = fv 2 \p>1W 2;p(Bb);
v 2 C (B b); Bv = 0 on @Bb g; where   @v N ? 1 b Bv = @n + b ? 2 v = 0 on @Bb (4.20)

must have unstable eigenvalues: one with a 1-dimensional eigenspace spanned by a radial eigenfunction corresponding to a shift in t, and another with an N -dimensional eigenspace corresponding to shifts in x1 ; x2 ; : : : ; xN . We will show below by means of power series developments that except for these two unstable eigenvalues, (L) consists entirely of negative eigenvalues. Therefore the unstable manifold provided by Theorem 1.3 consists only of the images of (4.17) under the transformation in Section 2. Every orbit in the unstable manifold has just the same selfsimilar pro le, disguised by the transformation to selfsimilar variables which depends on an arbitrary choice of the focussing point and time. This may be interpreted by saying that, although the equilibrium (4.16) is unstable, the pro le itself is stable. This is very much like the travelling wave case where the translates of the equilibrium form a center manifold. It remains to show that the spectrum of (4.18) with boundary condition (4.20) has the desired properties. As in the example in Section 4.1, the spectrum consists only of real eigenvalues which now are to be found from     00 + N ? 1 ? r 0 + 1 ? n(n + N ? 2) =  ; (r)  rn as r ! 0; (4.21) r 2 2 r2 with boundary condition   0 (b) + N ? 1 ? b (b) = 0: (4.22) b 2 16

Note that the solution of (4.21) is a multiple of the power series (using the Pochhammer symbols (k)j de ned by (k)0 = 1 and (k)j +1 = (k + j )(k)j ) (r) =

1 X

( + n?2 1 )j n+2j n  + n?2 1 n+2 =r + r + N r j 4(n + N2 ) j =0 4 j !(n + 2 )j

( + n?2 1 )( + n?2 1 + 1) n+4 ( + n?2 1 )( + n?2 1 + 1)( + n?2 1 + 2) n+6 r + 3 r +


(4.23) 42 2!(n + N2 )(n + N2 + 1) 4 3!(n + N2 )(n + N2 + 1)(n + N2 + 2) We number the eigenvalues again by (jn) (j = 1; 2 : : :) with corresponding solutions j(n) . The monotonicity properties of (jn) and the sign change properties of j(n) are the same as in Section 4.1, i.e. (1n) > (2n) > (3n) >


# ?1, j(n) has j ? 1 sign changes in the interval (0; b) and the double sequence (jn) is also decreasing in j . The eigenvalues due to shifts in the original variables are (1) 1 (1) (0) 0 00 N ? 1 0 (4.24) (0) 1 = 1 with 1 = f + r f and 1 = 2 with 1 = f : We have to show that these are the only nonnegative eigenvalues. In view of the mono(2) tonicity properties it suces to show that (0) 2 and 1 are negative. By the radial analysis in [3] we have that (0) 2 < 0. We recall that this is because f solves (4.21) with  = 0 and has a rst sign change at r = b. Therefore the solution of (4.21) with n = 0 and   0 cannot have a sign change before z = b. Since 2(0) must change sign, it follows that (0) 2 < 0. It remains to show that (2) 1 is negative and this is more involved. From here on let (2) n = 2. We will show that 1 must lie in the interval (?1; 0). To do so we rst observe that for   ?1 the function de ned by (4.23) is positive on (0; b]. Indeed we have for  = ?1 that 1 X (? 21 )j r2+2j ; (4.25) (r) = N j j =0 4 j !(2 + 2 )j whence, using 1 (? 1 )j X 2j 2 (4.26) g(b) = j j ! ( N )j b = 0 4 2 j =0 to simplify expressions, 1 (? 1 )j 1 X (? 12 )j 2+2j 2 X 2j 2 b ? b (b) = (b) ? b2 g(b) = j j !(2 + N )j j j !( N )j b > 0; 4 4 j =0 2 2 j =0 so that, because only the rst term in (4.25) has a positive coecient, (r) > 0 on (0; b]. Increasing  from  = ?1 we only make larger on (0; b] so we conclude that (r) > 0 on (0; b] for all   ?1. To nish we prove that (B )(b) goes from positive to negative when  goes from 0 to ?1. From (4.23) we derive that (B )(b) = (N + 1)b +

1 X

( + 21 )j ?1 ((2j + N + 1) ? 2j ? N 2+ 1 )b1+2j : N j j =1 4 j ! ( 2 + 2)j

Thus, when  = 0, we have, using (4.26), (B )(b) = (N + 1)b ?

1 X

( 21 )j ?1 (2j + N 2+ 1 )b1+2j ? (N + 1)bg(b) N j j =1 4 j ! ( 2 + 2)j 17

1 ( 1 )j ?1 N +1 X 2 2 j j! ( ( N ) 4 j =1 2 j

N +1

? 2(jN++ 2)2 )b2j = 2

j

For  = ?1 we have, using (4.26) again, (B )(b) = (N +1)b ?

1 X

( 12 )j ?1 Nj + j + 1 2j b > 0: N j j =1 4 (j ? 1)! 2( 2 )j +2

1 X

(? 21 )j ?1 N +1 )b1+2j +( N 2 +3N ? 2 b2 ?N ?1)bg(b) = (4 j +3 N j 2 2N (N + 4) j =1 4 j ! ( 2 + 2)j

1 4(N 2+3N?2)j 2 + 4(N+3)(N 2+2N?2)j +N 4+8N 3+15N 2?12N?16 (? 1 )j X 2 b1+2j N j +1 2 N ( N + 4)( j ? 1)! 4 ( 2 )j +3 j =1

< 0;

because (? 21 )j < 0 and the rst numerator in the latter expression is, setting j = 1, larger then (N + 4)(N 3 + 8N 2 + 7N ? 12) > 0.

References [1] H. Berestycki, L.A. Caarelli & L. Nirenberg, Uniform Estimates for Regularization of Free Boundary Problems, in: Analysis and Partial Dierential Equations, Vol. 122, C. Sadosky, Ed, Marcel Dekker, pp. 567-619, 1990. [2] M. Bertsch, D. Hilhorst & C. Schmidt-Laine, The well-posedness of a free-boundary problem arising in combustion theory, Nonl. Anal. 23, pp. 1211-1224, 1994. [3] C.M. Brauner, J. Hulshof & Cl. Schmidt-Laine, The saddle point property for focusing selfsimilar solutions of a combustion type free boundary problem: the radial case, Proc. AMS, 127. pp 473-479, 1999. [4] C.M. Brauner, A. Lunardi & Cl. Schmidt-Laine, Stability of travelling waves with interface conditions, Nonl. Anal. 19, pp. 455-474, 1992. [5] C.M. Brauner, A. Lunardi & Cl. Schmidt-Laine, Multi-dimensional stability analysis of planar travelling waves, Appl. Math. Letters 7, pp. 1-4, 1994. [6] C.M. Brauner, A. Lunardi & Cl. Schmidt-Laine, Stability of travelling waves in a multidimensional free boundary problem, Preprint Scuola Norm. Sup. Pisa 3, 1996. [7] D. Bresch & J. Simon, Sur les variations normales d'un domaine, ESAIM: Control, Optimisation and Calculus of Variations, 3, pp. 251-261, 1998. [8] D. Bucur & J.P. Zolesio, Shape stability of eigenvalues under capacitary constraints, Journal of Convex Analysis, 5, pp. 19-30, 1998. [9] L.A. Caarelli & J.L. Vazquez, A free boundary problem for the heat equation arising in ame propagation, Trans. Amer. Math. Soc. 347, pp. 411-441, 1995. [10] A. Dervieux, Perturbation des equations d'equilibre d'un plasma con ne: comportement de la frontiere libre, etude des branches de solutions, INRIA Research Report 18, 1980. [11] V.A. Galaktionov, J. Hulshof & J.L. Vazquez, Extinction and focusing behaviour of spherical and annular ames described by a free boundary problem, Journal Math. Pures et Appl., 76, pp. 563-608, 1997. [12] J. Hadamard, Memoire sur le probleme d'analyse relatif a l'equilibre des plaques elastiques encastrees, in: Oeuvres de J. Hadamard, 2 (1907) CNRS, Paris 1968. 18

[13] D. Hilhorst & J. Hulshof, An elliptic-parabolic problem in combustion theory: convergence to travelling waves, Nonl. Anal. 17, pp. 519-546, 1991. [14] D. Hilhorst & J. Hulshof, A free boundary focusing problem, Proc. AMS 121, pp. 1193-1202, 1994 [15] O.A. Ladyzhenskaja, V.A. Solonnikov, N.N. Ural'ceva, Linear and quasilinear equations of parabolic type, Nauka, Moskow 1967 (Russian). English transl.: Transl. Math. Monographs, AMS, Providence (1968). [16] J.D. Buckmaster & G.S.S. Ludford, Theory of Laminar Flames, Cambridge University Press, Cambridge, 1982. [17] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, Basel, 1995. [18] J. Simon, Dierentiation with respect to the domain in boundary value problems, Numer. Funct. Optimiz. 2, pp. 649-687, 1980. [19] J. Simon, Optimum design for Neumann condition and for related boundary value problems, in: Boundary control and boundary variations, J.P. Zolesio, Ed., Lecture Notes in Control and Information Sciences 100, Springer, 1988. [20] H.B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc. 259 , pp. 299-310, 1980. [21] J.L. Vazquez, The free boundary problem for the heat equation with xed gradient condition, Proc. Int. Conf. `Free Boundary Problems, Theory and Applications', Zakopane, Poland, Pitman Research Notes in Mathematics 363, pp. 277-302, 1995.

19

Appendix: Asymptotic behavior in linear problems Throughout the appendix we use the notation of Section 3. Let be a bounded open set in RN with C 2+ boundary, 0 <  < 1. Consider the linear problem 8 ut = Lw + f (; t); t  0;  2 ; > > > > < > > > > :

Bw = g(; t); t  0;  2 @ ;

(0.1)

u(; 0) = u0( );  2 ;

where the elliptic operator L and the nontangential operator B satisfy assumption H3, and u0 2 C 2+ ( ). The realisation L of L with homogeneous boundary conditions in X = C ( ), de ned in (3.5), is a sectorial operator thanks to [20]. Moreover if f , g and u0 are regular enough the unique solution of (0.1) is given by the extension of the Balakrishnan formula (see e.g. [17, p.200]) u(
; t) = etL (u0 ? G(
; 0)) + Zt

Z t

0

e(t?s)L[f (
; s) + LG(
; s)]ds

?L e(t?s)L[G(
; s) ? G(
; 0)]ds + G(
; 0) 0

= etL u0 +

Z t

0

e(t?s)L [f (
; s) + LG(
; s)]ds ? L

(0.2) Zt

0

e(t?s)LG(
; s)ds; 0  t  T:

Here G(
; t) = N g(
; t), the operator N being any lifting operator such that 8
0: ? t

kLe k P ? etLekL(X )  Cktek ; kLe k (I ? P ? )e?tLekL(X )  Ck e? t; t > 0; k 2 N:

(0.6)

P It is convenient to split v(t) = v(
; t) as v = 3i=1 vi , where

v1 (t) = etLeP ?u0 ? (etLe ? I )P ? N ge(
; 0)

+

Z t

0

e ge(
; s)]ds ? e(t?s)LeP ? [fe(
; s) + LN

v2 (t) = ?Le

Z t

v3 (t) = +Le

t

e ge]ds; e(t?s)Le(I ? P ? )[fe + LN

e(t?s)LeP ? N [ge(
; s) ? ge(
; 0)]ds;

0 Z +1

t

Z +1

e(t?s)Le(I ? P ? )N ge(
; s)ds:

e ge(
; t) The function v1 may be estimated arguing as in [17]. In fact, both t ! fe(
; t) and t ! LN = 2  belong to C ([0; 1); X ) \ B ([0; 1); C ( )). Therefore,

kv1 kC 1+=2([1;1);X ) + kv1 kB([1;1);DL(1+=2;1))  C (ku0 kX + kfek + kgek): Let us consider v2 . Since t ! ge(
; t) 2 C 1=2+=2 ([0; 1); C (@ )) we have N ge 2 C 1=2+=2 ([0; 1); Moreover C 1 ( ) is continuously embedded in DL(1=2; 1), see e.g. [17, Thm. 3.1.31]. Applying [17, Thm. 4.3.16] with  = 1=2 + =2, = =2, we get v2 2 C 1+=2 ([0; T ]; X ) and C 1 ( )).

21

t ! v2 (t) ? (I ? P )(N ge(
; t) ? N ge(
; 0)) 2 B ([0; T ]; DL(1 + =2; 1)) for every T > 0. Looking at the proof of Theorem 4.3.16 and of the previous Theorem 4.3.1(iii) one sees that

[v20 ]C =2 ([0;T ];X ) + sup0tT [L(v2 (t) ? P ? (N ge(t) ? N ge(0))]DL (=2;1)

 C kN gekC 1=2+=2([0;1);C 1( ))

;

whith constant C independent of T . A similar estimate holds for the lower order norms, as is easily seen using (0.6). Recalling that DL(1 + =2; 1) is continuously embedded in C 2+ ( ) (see [17, Thm. 3.1.34(ii)]) and that P ?N ge is bounded with values in C 2+ ( ), we get v2 2 C 1+=2 ([1; 1); X ) \ B ([1; 1); C 2+ ( )) and

kv2 kC 1+=2([1;1);X ) + kv2 kB([1;1);C 2+( ))  C kgek: Finally let us consider v3 . By estimates (0.6) it is obviously bounded with values in D(Lk ) for e 3?L e (I ? P ? )N ge is Holder continuous with values in X and every k 2 N . Moreover v30 = Lv

kv3 kC 1+=2([1;1);X ) + kv3 kB([1;1);DL(1+=2;1))  C kgek: Summing up and recalling once again that DL (1 + =2; 1)  C 2+ ( ) we get

kvkC 1+=2([1;1);X ) + kvkB([1;1);C 2+( ))  C (ku0 kX + kfek + kgek): It follows that v 2 C 2+;1+=2 (  [1; 1)), and

kvkC 2+;1+=2( [1;1))  C (ku0 kX + kfek + kgek); which nishes the proof. Let us now consider the backward problem

8 ut = Lu + f (; t); t  0;  2 ; > > > > < > > > > :

Bu = g(; t); t  0;  2 @ ;

(0.7)

u(; 0) = u0 ( );  2 :

Theorem 0.2 Let + (L) 6= ; and 0 < ! < minfRe :  2 +(L)g. Let f be such that (; t) ! e?!t f (; t) 2 C ;=2 (  (?1; 0]) and let g be such that (; t) ! e?!t g(; t) 2 C 1+;1=2+=2 (@  (?1; 0]), u0 2 C 2+ ( ). Then Problem (0.7) has a solution u such that v(; t) = e?!tu(; t) is bounded in (?1; 0] 

if and only if

(I ? P + )u0 =

Z 0

?1

e?sL (I ? P + )[f (
; s) + LN g(
; s)]ds ? L

Z0

In this case, u is given by u(
; t) = etLP + u0 +

+

Zt

?1

Z t

0

e(t?s)L P + [f (
; s) + LN g(
; s)]ds ? L

e(t?s)L(I ? P + )[f (
; s) + LN g(
; s)]ds ? L

Z t

?1

?1 Zt

0

e?sA (I ? P + )N g(
; s)ds: (0.8)

e(t?s)LP + N g(
; s)ds

e(t?s)L (I ? P + )N g(
; s)ds; t  0:

Moreover, v belongs to C 2+;1+=2 (  (?1; 0]) and

kvkC 2+;1+=2( (?1;0])  C (ku0 kC ( ) + ke?!tf kC ;=2( (?1;0]) + ke?!t gkC 1+;1=2+=2(@ (?1;0]) ): 22

(0.9)

Proof | Let Le = L ? !I . For every > 0 we have

? t

kLe k P + e?tLekL(X )  Ck e? t ; kLe k (I ? P + )etLekL(X )  Ck e tk ; t > 0; k 2 N:

(0.10)

Using (0.10) it is not hard to see that, if u is the function de ned by (0.9), v(t) = e?!tu(
; t) is bounded. Proving that condition (0.8) is necessary for v to be bounded is similar to [17, Thm. 4.4.6] and is left to the reader. Concerning P + v, we remark that it satis es (see [17, Thm. 5.1.18]) 8 + 0 e + v (t) + P + (fe(
; t) ? LN e ge(
; t)); t  0; < (P v ) (t) = LP :

P + v(0) = P + u0 ;

with fe = fe!t and ge = ge!t , so that P + v(t) = etLeP + u0 +

Z t

0

e ge]ds ? L e e(t?s)LeP + [fe + LN

Z t

0

e(t?s)L P +N geds; t  0;

and (0.9) holds. Let us prove that v 2 C 1+=2 ((?1; 0]; X ) \ B ((?1; 0]; C 2+ ( )). Using estimates (0.10) it is easy to see that P + v is bounded in (?1; 0] with values in D(Le k ) for every k 2 N . Moreover, e ge(
; t) are in C =2 ((?1; 0]; X ) (see the proof of Theorem 0.1), we since t ! fe(
; t) and t ! LN + 0 = 2 have P v 2 C ((?1; 0]; X ). It follows that P + v 2 C 1+=2 ((?1; 0]; X ) \ B ((?1; 0]; DL(1 + =2; 1))  C 1+=2 ((?1; 0]; X ) \ B ((?1; 0]; C 2+ ( )) and

kP +vkC 1+=2 ((?1;0];X ) + kP +vkB((?1;0];C 2+( ))  C (ku0 kX + kfek + kgek): Let us split (I ? P + )v as (I ? P + )v = v1 + v2 , where v1 (t) =

Z t

?1

e ge]ds; t  0; e(t?s)Le(I ? P + )[fe + LN Z t

v2 (t) = ?Le

?1

e(t?s)Le(I ? P + )N geds; t  0:

e ge(
; t) belong to C =2 ((?1; 0]; X ) \ B ((?1; 0]; DL (=2; 1)), Since t ! fe(
; t) and t ! LN by [17, Prop. 4.4.5(ii)(iii)] v1 belongs to C 1+=2 ((?1; 0]; X ) \ B ((?1; 0]; DL(1 + =2; 1))  C 1+=2 ((?1; 0]; X ) \ B ((?1; 0]; C 2+ ( )) and

kv1 kC 1+=2((?1;0];X ) + kv1 kB((?1;0];C 2+( ))  C (kfek + kgek): To estimate v2 we recall that N ge 2 C 1=2+=2 ((?1; 0]; C 1 ( ))  C 1=2+=2 ((?1; 0]; DL(1=2; 1)) (see again the proof of Theorem 0.1). By [17, Prop. 4.4.5(ii)] and [17, Prop. 2.2.12(i)], applied with X replaced by DL (1=2; 1), the function z (t) = ?

Z t

?1

e(t?s)Le(I ? P + )N ge(s)ds

is such that z 0 is bounded with values in DL(1 + =2; 1)  C 2+ ( ). Since z 0 = v2 ? (I ? P +)N ge, and (I ? P + )N ge is bounded with values in C 2+ ( ), we get v2 2 B ((?1; 0]; C 2+ ( )) and

kv2 kB((?1;0];C 2+( ))  C kgek: e (t), where Let now a < 0 and consider the restriction of v2 to [a; 0]. It is equal to Lw

w(t) = ?e(t?a)Le

= e(t?a)Lez (a) ?

Z a

?1

Z t

a

e(a?s)Le(I ? P + )N ge(s)ds ?

Z t

a

e(t?s)Le(I ? P + )N ge(s)ds

e(t?s)Le(I ? P +)N ge(s)ds; a  t  0:

23

e (a) ? (I ? P + )N ge(a;
) = z 0 (a) 2 DL (1 + =2; 1), with norm indeWe have just proved that Lz pendent of a. By [17, Thm 4.3.16] and the same arguments used in the proof of Theorem 0.1, we get v2 2 C 1+=2 ((?1; 0]; X ) and

kv2 kC 1+=2((?1;0];X )  C kgek: Summing up, we nd that v 2 C 1+=2 ((?1; 0]; X ) \ B ((?1; 0]; C 2+ ( )) so that v 2

C 2+;1+=2 (  (?1; 0]) and

kvkC 2+;1+=2( (?1;0])  C (ku0 kX + kfek + kgek); which nishes the proof.

24